Finite Group Databasebeta

← #26142841

Subgroups — #26142841

Order 12032.

1658 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689016C212yesmaximal, complemented
2#20241641C212yesmaximal, complemented
3#20241637C212yesmaximal, complemented
4#20241633C212yesmaximal, complemented
5#20241629C212yesmaximal
6#20241625C212yesmaximal, complemented
7#20241615C211yesmaximal, characteristic, complemented
8C4×C47⋊#95C211yesmaximal, characteristic, complemented
9C2×C47×#390C211yesmaximal, characteristic
10C22×C4⋊Dic47C2212yes—
11C22×C22⋊Dic47C2212yes—
12C22×C22⋊Dic47C2212yes—
13C22×C22⋊Dic47C2212yes—
14C22×C22⋊Dic47C2212yes—
15C22×C4×Dic47C2211yescharacteristic
16C22×C4×Dic47C2211yescharacteristic
17#15689016—28no—
18C2×C47⋊#95C2212yes—
19C2×C47⋊#95C2212yes—
20C2×C47⋊#95C2212yes—
21C2×C47⋊#95C2212yes—
22C2×C47⋊#95C2212yes—
23C2×C47⋊#95C2211yescharacteristic
24C2×C47⋊#95C2211yescharacteristic
25C2×C47⋊#95C2212yes—
26C2×C47⋊#95C2212yes—
27C23×D4×C47C2211yescharacteristic
28C22×C4⋊C4×C47C2211yescharacteristic
29C22×C22⋊C4×C47C2212yes—
30C22×C22⋊C4×C47C2212yes—
31C22×C42×C47C2211yescharacteristic
32C47×#390C418yescomplemented
33C23×Dic47C2312yes—
34C23×Dic47C2312yes—
35C23×Dic47C2312yes—
36C23×Dic47C2312yes—
37C2×C4⋊Dic47—24no—
38C2×C4⋊Dic47—42no—
39C2×C4⋊Dic47—42no—
40C2×C22⋊Dic47—24no—
41C2×C22⋊Dic47—24no—
42C2×C22⋊Dic47—24no—
43C2×C22⋊Dic47—24no—
44C2×C22⋊Dic47—42no—
45C2×C22⋊Dic47—42no—
46C2×C22⋊Dic47—42no—
47C2×C22⋊Dic47—42no—
48C2×C22⋊Dic47—42no—
49C2×C22⋊Dic47—42no—
50C2×C22⋊Dic47—42no—
51C2×C22⋊Dic47—42no—
52C2×C4×Dic47—22no—
53C2×C4×Dic47—41no—
54C2×C4×Dic47—41no—
55C2×C4×Dic47—41no—
56C2×C4×Dic47—41no—
57C2×C4×Dic47D414yescomplemented
58C47⋊#95—24no—
59C47⋊#95—24no—
60C47⋊#95—24no—
61C47⋊#95—24no—
62C47⋊#95—24no—
63C47⋊#95—42no—
64C47⋊#95—42no—
65C47⋊#95—41no—
66C47⋊#95—41no—
67C25×C47C2312yes—
68C22×D4×C47—24no—
69C22×D4×C47—41no—
70C22×D4×C47—41no—
71C22×D4×C47—42no—
72C22×D4×C47C2×C414yes—
73C23×C4×C47C2311yescharacteristic
74C23×C4×C47C2312yes—
75C23×C4×C47C2311yescharacteristic
76C23×C4×C47C2311yescharacteristic
77C2×C4⋊C4×C47—42no—
78C2×C4⋊C4×C47C2×C414yes—
79C2×C22⋊C4×C47—24no—
80C2×C22⋊C4×C47—24no—
81C2×C22⋊C4×C47—42no—
82C2×C22⋊C4×C47—42no—
83C2×C22⋊C4×C47C2×C418yes—
84C2×C22⋊C4×C47C2×C418yes—
85C2×C42×C47—41no—
86C2×C42×C47—41no—
87C2×C42×C47C2×C414yes—
88C24×C47—22no—
89C24×C47—22no—
90C24×C47—22no—
91C24×C47—22no—
92C24×C47—22no—
93C24×C47—22no—
94C24×C47—22no—
95C24×C47—22no—
96C24×C47—22no—
97C24×C47—22no—
98C24×C47—22no—
99C24×C47—22no—
100C24×C47—22no—
101C24×C47—22no—
102C24×C47C22×C414yes—
103C24×C47C2411yescharacteristic
104C2×D4×C47—28no—
105C2×D4×C47—44no—
106C2×D4×C47—44no—
107C2×D4×C47—48no—
108C2×D4×C47—48no—
109C22×C4×C47—21no—
110C22×C4×C47—21no—
111C22×C4×C47—21no—
112C22×C4×C47—21no—
113C22×C4×C47—22no—
114C22×C4×C47—21no—
115C22×C4×C47—21no—
116C22×C4×C47—22no—
117C22×C4×C47—22no—
118C22×C4×C47—22no—
119C22×C4×C47—21no—
120C22×C4×C47—21no—
121C22×C4×C47—22no—
122C22×C4×C47—22no—
123C22×C4×C47—22no—
124C22×C4×C47—22no—
125C22×C4×C47—22no—
126C22×C4×C47C22×C412yes—
127C22×C4×C47C22×C414yes—
128C22×C4×C47C22×C412yes—
129C22×C4×C47C22×C412yes—
130C22×C4×C47C2×D412yes—
131C22×C4×C47#4114yes—
132C22×C4×C47#4112yes—
133C22×Dic47—22no—
134C22×Dic47—22no—
135C22×Dic47—22no—
136C22×Dic47—22no—
137C22×Dic47—22no—
138C22×Dic47—22no—
139C22×Dic47—22no—
140C22×Dic47—22no—
141C22×Dic47—22no—
142C22×Dic47—22no—
143C22×Dic47—22no—
144C22×Dic47—22no—
145C22×Dic47—22no—
146C22×Dic47—22no—
147C22×Dic47—22no—
148C22×Dic47—22no—
149C22×Dic47—22no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47—22no—
153C22×Dic47—22no—
154C22×Dic47—22no—
155C22×Dic47—22no—
156C22×Dic47—22no—
157C22×Dic47—22no—
158C22×Dic47—22no—
159C22×Dic47C2×D414yes—
160C22×Dic47#4114yes—
161C4⋊C4×C47—44no—
162C22⋊C4×C47—216no—
163C22⋊C4×C47—48no—
164C42×C47—44no—
165C22⋊Dic47—48no—
166C22⋊Dic47—48no—
167C22⋊Dic47—48no—
168C22⋊Dic47—48no—
169C4⋊Dic47—48no—
170C4×Dic47—44no—
171C4×Dic47—44no—
172C23×C47—24no—
173C23×C47—24no—
174C23×C47—24no—
175C23×C47—24no—
176C23×C47—24no—
177C23×C47—24no—
178C23×C47—24no—
179C23×C47—42no—
180C23×C47—42no—
181C23×C47—42no—
182C23×C47—42no—
183C23×C47—42no—
184C23×C47—42no—
185C23×C47—42no—
186C23×C47—42no—
187C23×C47—42no—
188C23×C47—42no—
189C23×C47—42no—
190C23×C47—42no—
191C23×C47—42no—
192C23×C47—42no—
193C23×C47—42no—
194C23×C47—42no—
195C23×C47—42no—
196C23×C47—42no—
197C23×C47—42no—
198C23×C47—42no—
199C23×C47—42no—
200C23×C47—42no—
201C23×C47—42no—
202C23×C47—42no—
203C23×C47—42no—
204C23×C47—42no—
205C23×C47—42no—
206C23×C47—42no—
207C23×C47C23×C411yescharacteristic
208C23×C47C22×D411yescharacteristic
209C23×C47C2×#4112yes—
210C23×C47C2×#4111yescharacteristic
211C23×C47C2×#4112yes—
212C23×C47H(5,𝔽2)11yescharacteristic
213C23×C47H(5,𝔽2)12yes—
214C23×C47H(5,𝔽2)12yes—
215C23×C47H(5,𝔽2)11yescharacteristic
216C23×C47H(5,𝔽2)11yescharacteristic
217C23×C472−1+411yescharacteristic
218D4×C47—416no—
219C2×C4×C47—22no—
220C2×C4×C47—22no—
221C2×C4×C47—22no—
222C2×C4×C47—22no—
223C2×C4×C47—22no—
224C2×C4×C47—22no—
225C2×C4×C47—22no—
226C2×C4×C47—24no—
227C2×C4×C47—24no—
228C2×C4×C47—22no—
229C2×C4×C47—24no—
230C2×C4×C47—22no—
231C2×C4×C47—22no—
232C2×C4×C47—22no—
233C2×C4×C47—24no—
234C2×C4×C47—24no—
235C2×C4×C47—42no—
236C2×C4×C47—42no—
237C2×C4×C47—42no—
238C2×C4×C47—42no—
239C2×C4×C47—42no—
240C2×C4×C47—42no—
241C2×C4×C47#11714yes—
242C2×C4×C47C4×D414yes—
243C2×Dic47—24no—
244C2×Dic47—24no—
245C2×Dic47—24no—
246C2×Dic47—24no—
247C2×Dic47—24no—
248C2×Dic47—24no—
249C2×Dic47—42no—
250C2×Dic47—42no—
251C2×Dic47—42no—
252C2×Dic47—42no—
253C2×Dic47—42no—
254C2×Dic47—42no—
255C2×Dic47—42no—
256C2×Dic47—42no—
257C2×Dic47—42no—
258C2×Dic47—42no—
259C2×Dic47—42no—
260C2×Dic47—42no—
261C2×Dic47—42no—
262C2×Dic47—42no—
263C2×Dic47—42no—
264C2×Dic47—42no—
265C2×Dic47—42no—
266C2×Dic47—42no—
267C2×Dic47—42no—
268C2×Dic47—42no—
269C2×Dic47—42no—
270C2×Dic47—42no—
271#14278—471nomaximal, Sylow
272C22×C47—44no—
273C22×C47—44no—
274C22×C47—44no—
275C22×C47—44no—
276C22×C47—44no—
277C22×C47—44no—
278C22×C47—44no—
279C22×C47—82no—
280C22×C47—82no—
281C22×C47—82no—
282C22×C47—82no—
283C22×C47—82no—
284C22×C47—82no—
285C22×C47—82no—
286C22×C47—82no—
287C22×C47C2×#11711yescharacteristic
288C22×C47C2×C4×D411yescharacteristic
289C22×C47C4×#4111yescharacteristic
290C22×C47#51811yescharacteristic
291C22×C47#51811yescharacteristic
292C22×C47#51811yescharacteristic
293C22×C47#52011yescharacteristic
294C22×C47#52912yes—
295C22×C47#53212yes—
296C22×C47#53411yescharacteristic
297C22×C47#53512yes—
298C22×C47#53711yescharacteristic
299C22×C47#53812yes—
300C22×C47#53812yes—
301C22×C47#53811yescharacteristic
302C22×C47#53812yes—
303C22×C47#53812yes—
304C22×C47#53911yescharacteristic
305C22×C47#54012yes—
306C22×C47#54012yes—
307C22×C47#54012yes—
308C22×C47#54212yes—
309C22×C47#54212yes—
310C4×C47—24no—
311C4×C47—24no—
312C4×C47—44no—
313C4×C47—42no—
314Dic47—44no—
315Dic47—44no—
316Dic47—82no—
317Dic47—82no—
318C2×#390D4711yescharacteristic
319C2×#388—472no—
320#1441—472no—
321#1440—472no—
322#1439—472no—
323#1438—472no—
324#1437—472no—
325#1434—471no—
326C4×#95—471no—
327C2×C47—84no—
328C2×C47#229311yescharacteristic
329C2×C47#231211yescharacteristic
330C2×C47#231911yescharacteristic
331C2×C47#232511yescharacteristic
332C2×C47#232811yescharacteristic
333C2×C47#233311yescharacteristic
334C2×C47#233711yescharacteristic
335C2×C47#264812yes—
336C2×C47#265112yes—
337C2×C47#266112yes—
338C2×C47#266212yes—
339C23×D4C2×D4711yescharacteristic
340C22×C4⋊C4—472no—
341C22×C4⋊C4C2×D4711yescharacteristic
342C22×C22⋊C4—472no—
343C22×C22⋊C4—472no—
344C22×C22⋊C4—472no—
345C22×C22⋊C4—472no—
346C22×C22⋊C4C2×D4712yes—
347C22×C22⋊C4C2×D4712yes—
348C22×C42—471no—
349C22×C42—471no—
350C22×C42C2×D4711yescharacteristic
351#390Dic4718yescomplemented
352#388—948no—
353C2×#95—472no—
354C2×#95—472no—
355C2×#95—472no—
356C2×#95—472no—
357C2×#95—472no—
358C2×#95—471no—
359C2×#95—471no—
360C2×#95—472no—
361C2×#95—472no—
362C47#1427811yescharacteristic, complemented, Sylow
363C25C22×D4712yes—
364C22×D4—24no—
365C22×D4—41no—
366C22×D4—41no—
367C22×D4—42no—
368C22×D4C2×Dic4714yes—
369C23×C4—472no—
370C23×C4—472no—
371C23×C4—472no—
372C23×C4—472no—
373C23×C4C22×D4712yes—
374C23×C4C22×D4711yescharacteristic
375C23×C4C22×D4711yescharacteristic
376C23×C4C22×D4711yescharacteristic
377C2×C4⋊C4—42no—
378C2×C4⋊C4—944no—
379C2×C4⋊C4—1882no—
380C2×C4⋊C4—1882no—
381C2×C4⋊C4C2×Dic4714yes—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—24no—
384C2×C22⋊C4—42no—
385C2×C22⋊C4—42no—
386C2×C22⋊C4—944no—
387C2×C22⋊C4—944no—
388C2×C22⋊C4—944no—
389C2×C22⋊C4—944no—
390C2×C22⋊C4—1882no—
391C2×C22⋊C4—1882no—
392C2×C22⋊C4—1882no—
393C2×C22⋊C4—1882no—
394C2×C22⋊C4—1882no—
395C2×C22⋊C4—1882no—
396C2×C22⋊C4—1882no—
397C2×C22⋊C4—1882no—
398C2×C22⋊C4C2×Dic4718yes—
399C2×C22⋊C4C2×Dic4718yes—
400C2×C42—41no—
401C2×C42—41no—
402C2×C42—474no—
403C2×C42—942no—
404C2×C42—1881no—
405C2×C42—1881no—
406C2×C42—1881no—
407C2×C42—1881no—
408C2×C42C2×Dic4714yes—
409#95—944no—
410#95—944no—
411#95—944no—
412#95—944no—
413#95—944no—
414#95—1881no—
415#95—1881no—
416#95—1882no—
417#95—1882no—
418C24—22no—
419C24—22no—
420C24—22no—
421C24—22no—
422C24—22no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24C22×Dic4714yes—
433C24C23×D4711yescharacteristic
434C2×D4—28no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—48no—
438C2×D4—48no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—474no—
457C22×C4—474no—
458C22×C4—942no—
459C22×C4—942no—
460C22×C4—942no—
461C22×C4—942no—
462C22×C4—942no—
463C22×C4—942no—
464C22×C4—942no—
465C22×C4—942no—
466C22×C4—942no—
467C22×C4—942no—
468C22×C4—942no—
469C22×C4—942no—
470C22×C4—942no—
471C22×C4—942no—
472C22×C4—942no—
473C22×C4—942no—
474C22×C4—942no—
475C22×C4—942no—
476C22×C4—942no—
477C22×C4—942no—
478C22×C4—942no—
479C22×C4—942no—
480C22×C4—942no—
481C22×C4—942no—
482C22×C4—942no—
483C22×C4—942no—
484C22×C4#1062535014yes—
485C22×C4D4×D4712yes—
486C22×C4#1062535212yes—
487C22×C4C22×Dic4712yes—
488C22×C4C22×Dic4714yes—
489C22×C4C22×Dic4712yes—
490C22×C4C22×Dic4712yes—
491C4⋊C4—44no—
492C4⋊C4—1888no—
493C22⋊C4—216no—
494C22⋊C4—48no—
495C22⋊C4—1888no—
496C22⋊C4—1888no—
497C22⋊C4—1888no—
498C22⋊C4—1888no—
499C42—44no—
500C42—1884no—
501C42—1884no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23C23×Dic4711yescharacteristic
538C23C2×D4×D4711yescharacteristic
539C23#41×D4712yes—
540C23C2×#1062535211yescharacteristic
541C23C2×#1062535012yes—
542C23#1309406912yes—
543C23#1309406912yes—
544C23#1309406911yescharacteristic
545C23#1309406911yescharacteristic
546C23#1309407011yescharacteristic
547C23#1309407111yescharacteristic
548D4—416no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—944no—
572C2×C4—944no—
573C2×C4—944no—
574C2×C4—944no—
575C2×C4—944no—
576C2×C4—944no—
577C2×C4—1882no—
578C2×C4—1882no—
579C2×C4—1882no—
580C2×C4—1882no—
581C2×C4—1882no—
582C2×C4—1882no—
583C2×C4—1882no—
584C2×C4—1882no—
585C2×C4—1882no—
586C2×C4—1882no—
587C2×C4—1882no—
588C2×C4—1882no—
589C2×C4—1882no—
590C2×C4—1882no—
591C2×C4—1882no—
592C2×C4—1882no—
593C2×C4—1882no—
594C2×C4—1882no—
595C2×C4—1882no—
596C2×C4—1882no—
597C2×C4—1882no—
598C2×C4—1882no—
599C2×C4#1309398314yes—
600C2×C4D4×Dic4714yes—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22C2×#1309398311yescharacteristic
617C22C2×D4×Dic4711yescharacteristic
618C22#41×Dic4711yescharacteristic
619C22#1568959811yescharacteristic
620C22#1568959811yescharacteristic
621C22#1568959811yescharacteristic
622C22#1568960611yescharacteristic
623C22#1568965512yes—
624C22#1568967012yes—
625C22#1568968111yescharacteristic
626C22#1568968612yes—
627C22#1568970011yescharacteristic
628C22#1568970512yes—
629C22#1568970712yes—
630C22#1568971011yescharacteristic
631C22#1568971112yes—
632C22#1568971112yes—
633C22#1568971811yescharacteristic
634C22#1568972312yes—
635C22#1568972412yes—
636C22#1568972712yes—
637C22#1568973712yes—
638C22#1568973812yes—
639C4—24no—
640C4—24no—
641C4—44no—
642C4—42no—
643C4—3762no—
644C4—3762no—
645C4—1884no—
646C4—1884no—
647C2—84no—
648C2#2024585311yescharacteristic
649C2#2024598611yescharacteristic
650C2#2024603111yescharacteristic
651C2#2024609311yescharacteristic
652C2#2024611411yescharacteristic
653C2#2024615511yescharacteristic
654C2#2024620311yescharacteristic
655C2#2024938012yes—
656C2#2024942512yes—
657C2#2024951812yes—
658C2#2024952912yes—
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC2×C47×#390C2
socle, cosocleC24×C47C24
prime radicalC23×D4×C47C22
hypercenterC2×#390D47
hypocenterC47#14278
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C2×#390

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#15689016C2semidirect
2. #20241641C2semidirect
3. #20241637C2semidirect
4. #20241633C2semidirect
6. #20241625C2semidirect
7. #20241615C2semidirect
8. C4×C47⋊#95C2semidirect
32. C47×#390C4semidirect
57. C2×C4×Dic47D4semidirect
351. #390Dic47semidirect
362. C47#14278semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#156890162
#202416412
#202416372
#202416332
#202416292
#202416252
#202416152
C4×C47⋊#952
C2×C47×#3902
#1427847